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PMMH preserves the exact parameter posterior

Statement

Under the hypotheses of Particle marginal Metropolis–Hastings, assume the unbiasedness identity there holds and the evidence $C$ is finite and positive. Then $\tilde{π}$ is a probability measure whose parameter marginal is $π_{y}$ . With the terminal index included in $U$ , the joint marginal of $(θ, X_{0 : T}^{K})$ is the posterior $C^{- 1} π_{0} (d θ) γ_{θ, T} (d x_{0 : T})$ . The PMMH transition kernel is reversible, hence invariant, for $\tilde{π}$ ; its parameter and selected-path marginals are therefore exact for every fixed $N \geq 1$ . This is an invariance result only; it asserts no irreducibility, uniqueness, convergence rate, or mixing speed.

Facts & Assumptions

Given: The hypotheses of Particle marginal Metropolis–Hastings, including the finite positive evidence, the unbiased particle likelihood estimate, and retention of the full auxiliary state after rejection.

[A1]

The Axiom of Choice: the path Radon–Nikodym suppliers assume AC.

[F1]

Particle marginal Metropolis–Hastings: $\tilde{π} (d θ, d u) = C^{- 1} π_{0} (d θ) m_{θ} (d u) {\hat{L}}_{θ} (u)$ , and the proposal on $(θ, u)$ is $R (θ, d θ^{'}) m_{θ^{'}} (d u^{'})$ .

[F2]

The sequential particle estimator is unbiased for Feynman–Kac path integrals: for each nonnegative measurable path test function $φ$ with finite target integral, $E [{\hat{Z}}_{T} {\hat{π}}_{T}^{N} (φ)] = γ_{θ, T} (φ)$ ; in particular $E [{\hat{Z}}_{T}] = L_{θ} (y_{1 : T})$ .

[F3]

Metropolis–Hastings kernel: the general accepted-flow construction defines the accept/reject transition rule and retains rejected proposals as self-transitions.

[F4]

Metropolis–Hastings flow is reversible and invariant: that rule is a probability kernel, reversible and invariant for its probability target.

[F5]

Invariant and stationary distribution for a Markov kernel: invariance means equality of the target measure after integrating the transition kernel.

Proof

Given: [A1] and the hypotheses in the Statement.

1.1F1F2givenalgebra

Integrating [F1] first in $u$ gives $C^{- 1} π_{0} (d θ) E_{m_{θ}} {\hat{L}}_{θ} = C^{- 1} L_{θ} π_{0} (d θ)$ by [F2]. Its total mass is $C^{- 1} \int L_{θ} d π_{0} = 1$ , so $\tilde{π}$ is a probability and its parameter marginal is $π_{y}$ .

2.1F1F2step 1.1algebra

For a measurable set $B$ of parameter-path pairs, summing over the terminal index gives $\tilde{π} {(θ, X_{0 : T}^{K}) \in B} = C^{- 1} \int π_{0} (d θ) E ⁣ [{\hat{Z}}_{T} \sum_{k = 1}^{N} W_{T}^{k} 1_{B} (θ, X_{0 : T}^{k})] = C^{- 1} \int π_{0} (d θ) γ_{θ, T} (B_{θ})$ , because the sum is the weighted empirical path integral in [F2]. Thus the selected path marginal is the joint posterior.

2.2A1F1F3F4step 1.1construct

The proposal in [F1] is a probability kernel because it is the composition of the probability kernel $R$ and the auxiliary probability kernels $m_{θ^{'}}$ . Applying [F3] to this target and proposal defines the PMMH accept/reject rule, which is a probability kernel by [F4]; in the common-density case the factors $m_{θ} (u)$ cancel between the target and proposal ratios, leaving exactly the displayed ratio in Particle marginal Metropolis–Hastings. On rejection, [F3] retains the current extended state.

3.1F1F4F5step 1.1step 2.1∎

By [F4], this extended-space kernel is reversible and invariant for $\tilde{π}$ . Applying the measurable coordinate projections $(θ, u) \mapsto θ$ and $(θ, u) \mapsto (θ, X_{0 : T}^{K})$ to the invariant law gives the exact marginals in steps 1.1 and 2.1, in the sense of [F5].

Depends on

Used by

Refreshing the current estimate can destroy exactness Counterexample
Pseudo-marginal rejection retains the current likelihood estimate Example

Cited to discharge well-definedness by Particle marginal Metropolis–Hastings.

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Sources

Andrieu, Doucet, and Holenstein, Particle Markov chain Monte Carlo methods, §4.4 and Theorem 4 (standard reference, not scraped)
Särkkä and Svensson, Bayesian Filtering and Smoothing, 2nd ed., §16.3.5 (standard reference, not scraped)